Software Health Management (SWHM) is an emerging field which addresses the critical need to detect, diagnose, predict, and mitigate adverse events due to software faults and failures. These faults could arise for numerous reasons including coding errors, unanticipated faults or failures in hardware, or problematic interactions with the external environment. This paper demonstrates a novel approach to software health management based on a rigorous Bayesian formulation that monitors the behavior of software and operating system, performs probabilistic diagnosis, and provides information about the most likely root causes of a failure or software problem. Translation of the Bayesian network model into an efficient data structure, an arithmetic circuit, makes it possible to perform SWHM on resource-restricted embedded computing platforms as found in aircraft, unmanned aircraft, or satellites. SWHM is especially important for safety critical systems such as aircraft control systems. In this paper, we demonstrate our Bayesian SWHM system on three realistic scenarios from an aircraft control system: (1) aircraft file-system based faults, (2) signal handling faults, and (3) navigation faults due to IMU (inertial measurement unit) failure or compromised GPS (Global Positioning System) integrity. We show that the method successfully detects and diagnoses faults in these scenarios. We also discuss the importance of verification and validation of SWHM systems.